3.128 \(\int x^3 (a+a \sin (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=337 \[ \frac{8 a x^2 \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f^2}+\frac{16 a x^2 \sqrt{a \sin (e+f x)+a}}{f^2}-\frac{64 a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{27 f^4}-\frac{1280 a \sqrt{a \sin (e+f x)+a}}{9 f^4}+\frac{32 a x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^3}+\frac{640 a x \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^3}-\frac{4 a x^3 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f} \]

[Out]

(-1280*a*Sqrt[a + a*Sin[e + f*x]])/(9*f^4) + (16*a*x^2*Sqrt[a + a*Sin[e + f*x]])/f^2 + (640*a*x*Cot[e/2 + Pi/4
+ (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (8*a*x^3*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(
3*f) + (32*a*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (4*a*x^
3*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (64*a*Sin[e/2 + Pi/4 +
(f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(27*f^4) + (8*a*x^2*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]]
)/(3*f^2)

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Rubi [A]  time = 0.230083, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3319, 3311, 3296, 2638, 3310} \[ \frac{8 a x^2 \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f^2}+\frac{16 a x^2 \sqrt{a \sin (e+f x)+a}}{f^2}-\frac{64 a \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{27 f^4}-\frac{1280 a \sqrt{a \sin (e+f x)+a}}{9 f^4}+\frac{32 a x \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^3}+\frac{640 a x \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^3}-\frac{4 a x^3 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x^3 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(-1280*a*Sqrt[a + a*Sin[e + f*x]])/(9*f^4) + (16*a*x^2*Sqrt[a + a*Sin[e + f*x]])/f^2 + (640*a*x*Cot[e/2 + Pi/4
+ (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (8*a*x^3*Cot[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(
3*f) + (32*a*x*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(9*f^3) - (4*a*x^
3*Cos[e/2 + Pi/4 + (f*x)/2]*Sin[e/2 + Pi/4 + (f*x)/2]*Sqrt[a + a*Sin[e + f*x]])/(3*f) - (64*a*Sin[e/2 + Pi/4 +
(f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]])/(27*f^4) + (8*a*x^2*Sin[e/2 + Pi/4 + (f*x)/2]^2*Sqrt[a + a*Sin[e + f*x]]
)/(3*f^2)

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rubi steps

\begin{align*} \int x^3 (a+a \sin (e+f x))^{3/2} \, dx &=\left (2 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x^3 \sin ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx\\ &=-\frac{4 a x^3 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{8 a x^2 \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f^2}+\frac{1}{3} \left (4 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x^3 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx-\frac{\left (16 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x \sin ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{3 f^2}\\ &=-\frac{8 a x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{32 a x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{4 a x^3 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{64 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{27 f^4}+\frac{8 a x^2 \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f^2}-\frac{\left (32 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{9 f^2}+\frac{\left (8 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x^2 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{f}\\ &=\frac{16 a x^2 \sqrt{a+a \sin (e+f x)}}{f^2}+\frac{64 a x \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{8 a x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{32 a x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{4 a x^3 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{64 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{27 f^4}+\frac{8 a x^2 \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f^2}-\frac{\left (64 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{9 f^3}-\frac{\left (32 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x \cos \left (\frac{e}{2}-\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{f^2}\\ &=-\frac{128 a \sqrt{a+a \sin (e+f x)}}{9 f^4}+\frac{16 a x^2 \sqrt{a+a \sin (e+f x)}}{f^2}+\frac{640 a x \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{8 a x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{32 a x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{4 a x^3 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{64 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{27 f^4}+\frac{8 a x^2 \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f^2}-\frac{\left (64 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{f^3}\\ &=-\frac{1280 a \sqrt{a+a \sin (e+f x)}}{9 f^4}+\frac{16 a x^2 \sqrt{a+a \sin (e+f x)}}{f^2}+\frac{640 a x \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{8 a x^3 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{32 a x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{4 a x^3 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{64 a \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{27 f^4}+\frac{8 a x^2 \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f^2}\\ \end{align*}

Mathematica [A]  time = 1.11269, size = 231, normalized size = 0.69 \[ \frac{2 a \sqrt{a (\sin (e+f x)+1)} \left (-\frac{2 \left (\sin \left (\frac{e}{2}\right ) \left (-18 f^3 x^3-117 f^2 x^2+480 f x+968\right )+\cos \left (\frac{e}{2}\right ) \left (18 f^3 x^3-117 f^2 x^2-480 f x+968\right )\right )}{\sin \left (\frac{e}{2}\right )+\cos \left (\frac{e}{2}\right )}-\cos (f x) \left (2 \sin (e) \left (8-9 f^2 x^2\right )+3 f x \cos (e) \left (3 f^2 x^2-8\right )\right )+\sin (f x) \left (3 f x \sin (e) \left (3 f^2 x^2-8\right )+2 \cos (e) \left (9 f^2 x^2-8\right )\right )+\frac{24 f x \left (3 f^2 x^2-80\right ) \sin \left (\frac{f x}{2}\right )}{\left (\sin \left (\frac{e}{2}\right )+\cos \left (\frac{e}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}\right )}{27 f^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(2*a*((-2*((968 - 480*f*x - 117*f^2*x^2 + 18*f^3*x^3)*Cos[e/2] + (968 + 480*f*x - 117*f^2*x^2 - 18*f^3*x^3)*Si
n[e/2]))/(Cos[e/2] + Sin[e/2]) - Cos[f*x]*(3*f*x*(-8 + 3*f^2*x^2)*Cos[e] + 2*(8 - 9*f^2*x^2)*Sin[e]) + (2*(-8
+ 9*f^2*x^2)*Cos[e] + 3*f*x*(-8 + 3*f^2*x^2)*Sin[e])*Sin[f*x] + (24*f*x*(-80 + 3*f^2*x^2)*Sin[(f*x)/2])/((Cos[
e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))*Sqrt[a*(1 + Sin[e + f*x])])/(27*f^4)

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+a*sin(f*x+e))^(3/2),x)

[Out]

int(x^3*(a+a*sin(f*x+e))^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)*x^3, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+a*sin(f*x+e))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)*x^3, x)